Justin Whitehouse, Morgane Austern, Vasilis Syrgkanis
arXiv 15 Jul 2025 · Econometrics
arXiv:2507.11780 · PDF · DOI · OpenAlex · Extracted main text
Constructing confidence intervals for the value of an optimal treatment policy is an important problem in causal inference. Insight into the optimal policy value can guide the development of reward-maximizing, individualized treatment regimes. However, because the functional that defines the optimal value is non-differentiable, standard semi-parametric approaches for performing inference fail to be directly applicable. Existing approaches for handling this non-differentiability fall roughly into two camps. In one camp are estimators based on constructing smooth approximations of the optimal value. These approaches are computationally lightweight, but typically place unrealistic parametric assumptions on outcome regressions. In another camp are approaches that directly de-bias the non-smooth objective. These approaches don't place parametric assumptions on nuisance functions, but they either require the computation of intractably-many nuisance estimates, assume unrealistic $L^\infty$ nuisance convergence rates, or make strong margin assumptions that prohibit non-response to a treatment. In this paper, we revisit the problem of constructing smooth approximations of non-differentiable functionals. By carefully controlling first-order bias and second-order remainders, we show that a softmax smoothing-based estimator can be used to estimate parameters that are specified as a maximum of scores involving nuisance components. In particular, this includes the value of the optimal treatment policy as a special case. Our estimator obtains $\sqrt{n}$ convergence rates, avoids parametric restrictions/unrealistic margin assumptions, and is often statistically efficient.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Luedtke, Alexander R and Van Der Laan, Mark J (2016) Statistical inference for the mean outcome under a possibly non-unique optimal treatment strategy | 1.000 | 12 | 4 | 100% |
| 2 | Levis, Alexander W and Bonvini, Matteo and Zeng, Zhenghao and Keele,… (2023) Covariate-assisted bounds on causal effects with instrumental variables | 1.000 | 10 | 3 | 100% |
| 3 | Shi, Chengchun and Lu, Wenbin and Song, Rui (2020) Breaking the Curse of Nonregularity with Subagging–-Inference of the Mean Outcome under Optimal Treatment Regimes | 1.000 | 8 | 4 | 100% |
| 4 | Chernozhukov, Victor and Chetverikov, Denis and Demirer, Mert and Du… (2018) Double/debiased machine learning for treatment and structural parameters | 1.000 | 6 | 4 | 100% |
| 5 | Gupta, Chirag (2022) Post-hoc calibration without distributional assumptions | 1.000 | 6 | 3 | 100% |
| 6 | Laber, Eric B and Lizotte, Daniel J and Qian, Min and Pelham, Willia… (2014) Dynamic treatment regimes: Technical challenges and applications | 0.928 | 4 | 3 | 100% |
| 7 | Semenova, Vira (2023) Aggregated Intersection Bounds and Aggregated Minimax Values | 0.928 | 4 | 3 | 100% |
| 8 | Hirano, Keisuke and Porter, Jack R (2012) Impossibility results for nondifferentiable functionals | 0.843 | 3 | 3 | 100% |
| 9 | Goldberg, Yair and Song, Rui and Zeng, Donglin and Kosorok, Michael R (2014) Comment on “Dynamic treatment regimes: Technical challenges and applications” | 0.811 | 4 | 2 | 100% |
| 10 | Van Der Laan, Mark J and Rubin, Daniel (2006) Targeted maximum likelihood learning | 0.737 | 3 | 2 | 100% |
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